Rút gọn các biểu thức sau:
a) \(A = \frac{{{x^5}{y^{ - 2}}}}{{{x^3}y}}\,\,\,\left( {x,y \ne 0} \right);\) b) \(B = \frac{{{x^2}{y^{ - 3}}}}{{{{\left( {{x^{ - 1}}{y^4}} \right)}^{ - 3}}}}\,\,\,\left( {x,y \ne 0} \right).\)
a: \(A=\dfrac{x^5}{x^3}\cdot\dfrac{y^{-2}}{y}=x^2\cdot y^{-1}=\dfrac{x^2}{y}\)
b: \(B=\dfrac{x^2\cdot y^{-3}}{x^3\cdot y^{-12}}=\dfrac{x^2}{x^3}\cdot\dfrac{y^{-3}}{y^{-12}}=\dfrac{1}{x}\cdot y^{-3+12}=\dfrac{y^9}{x}\)
a) \(A=\dfrac{x^5y^{-2}}{x^3y}=\dfrac{x^5}{x^3}.\dfrac{1}{y^{2-1}}=x^{5-3}y^{-1}=x^2y^{-1}\).
b) \(B=\dfrac{x^2y^{-3}}{\left(x^{-1}y^4\right)^{-3}}=\dfrac{x^2y^{-3}}{x^3y^{-12}}=x^{2-3}y^{-3-\left(-12\right)}=\dfrac{1}{xy^9}\)
Cho x, y là các số thực dương. Rút gọn các biểu thức sau:
a) \(A = \frac{{{x^{\frac{1}{3}}}\sqrt y + {y^{\frac{1}{3}}}\sqrt x }}{{\sqrt[6]{x} + \sqrt[6]{y}}};\)
b) \(B = {\left( {\frac{{{x^{\sqrt 3 }}}}{{{y^{\sqrt 3 - 1}}}}} \right)^{\sqrt 3 + 1}}.\frac{{{x^{ - \sqrt 3 - 1}}}}{{{y^{ - 2}}}}.\)
a: \(A=\dfrac{x^{\dfrac{1}{3}}\cdot y^{\dfrac{1}{2}}+y^{\dfrac{1}{3}}\cdot x^{\dfrac{1}{2}}}{x^{\dfrac{1}{6}}+y^{\dfrac{1}{6}}}=\dfrac{x^{\dfrac{1}{3}}\cdot y^{\dfrac{1}{3}}\left(x^{\dfrac{1}{6}}+y^{\dfrac{1}{6}}\right)}{x^{\dfrac{1}{6}}+y^{\dfrac{1}{6}}}=x^{\dfrac{1}{3}}\cdot y^{\dfrac{1}{3}}=\left(xy\right)^{\dfrac{1}{3}}\)
b: \(B=\dfrac{x^{3+\sqrt{3}}}{y^2}\cdot\dfrac{x^{-\sqrt{3}-1}}{y^{-2}}=\dfrac{x^{3+\sqrt{3}-\sqrt{3}-1}}{y^{2-2}}=x^2\)
Rút gọn biểu thức: \(A = \frac{{{x^{\frac{3}{2}}}y + x{y^{\frac{3}{2}}}}}{{\sqrt x + \sqrt y }}\,\,\,\left( {x,y > 0} \right).\)
\(=\dfrac{xy\left(x^{\dfrac{1}{2}}+y^{\dfrac{1}{2}}\right)}{x^{\dfrac{1}{2}}+y^{\dfrac{1}{2}}}=xy\)
\(A=\dfrac{x^{\dfrac{3}{2}}y+xy^{\dfrac{3}{2}}}{\sqrt{x}+\sqrt{y}}=\left(x+y\right).\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\).
Rút gọn các biểu thức sau:
a) A=\(\dfrac{x\sqrt{y}+y\sqrt{x}}{x+2\sqrt{xy}+y}\)(x≥0 , y≥0 , xy≠0)
b) B=\(\dfrac{x\sqrt{y}-y\sqrt{x}}{x-2\sqrt{xy}+y}\)(x≥0 , y≥0 , x≠y)
c) C=\(\dfrac{3\sqrt{a}-2a-1}{4a-4\sqrt{a}+1}\)(a≥0 , a≠\(\dfrac{1}{4}\))
d) D=\(\dfrac{a+4\sqrt{a}+4}{\sqrt{a}+2}+\dfrac{4-a}{\sqrt{a}-2}\)(a≥0 , a≠4)
a) \(A=\dfrac{x\sqrt{y}+y\sqrt{x}}{x+2\sqrt{xy}+y}\)
\(A=\dfrac{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}{\left(\sqrt{x}+\sqrt{y}\right)^2}\)
\(A=\dfrac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)
b) \(B=\dfrac{x\sqrt{y}-y\sqrt{x}}{x-2\sqrt{xy}+y}\)
\(B=\dfrac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{y}\right)^2}\)
\(B=\dfrac{\sqrt{xy}}{\sqrt{x}-\sqrt{y}}\)
c) \(C=\dfrac{3\sqrt{a}-2a-1}{4a-4\sqrt{a}+1}\)
\(C=\dfrac{-\left(2a-3\sqrt{a}+1\right)}{\left(2\sqrt{a}\right)^2-2\sqrt{a}\cdot2\cdot1+1^2}\)
\(C=\dfrac{-\left(\sqrt{a}-1\right)\left(2\sqrt{a}-1\right)}{\left(2\sqrt{a}-1\right)^2}\)
\(C=\dfrac{-\sqrt{a}+1}{2\sqrt{a}-1}\)
d) \(D=\dfrac{a+4\sqrt{a}+4}{\sqrt{a}+2}+\dfrac{4-a}{\sqrt{a}-2}\)
\(D=\dfrac{\left(\sqrt{a}+2\right)^2}{\sqrt{a}+2}+\dfrac{\left(2-\sqrt{a}\right)\left(2+\sqrt{a}\right)}{\sqrt{a}-2}\)
\(D=\sqrt{a}+2-\dfrac{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}{\sqrt{a}-2}\)
\(D=\left(\sqrt{a}+2\right)-\left(\sqrt{a}+2\right)\)
\(D=0\)
Kết quả rút gọn biểu thức A=\(\frac{y}{x}\sqrt{\frac{x^2}{y^4}}\)với x>0,y>0
\(A=\frac{y}{x}\cdot\sqrt{\frac{x^2}{y^4}}=\frac{y}{x}\cdot\frac{\sqrt{x^2}}{\sqrt{y^4}}=\frac{y}{x}\cdot\frac{\left|x\right|}{\left|y^2\right|}=\frac{y}{x}\cdot\frac{x}{y^2}=\frac{1}{y}\)( x > 0 ; y > 0 )
Đề bài
Cho x; y là các số thực dương. Rút gọn mỗi biểu thức sau:
\(A = \frac{{{x^{\frac{5}{4}}}y + x.{y^{\frac{5}{4}}}}}{{\sqrt[4]{x} + \sqrt[4]{y}}}\)
\(B = {\left( {\sqrt[7]{{\frac{x}{y}\sqrt[5]{{\frac{y}{x}}}}}} \right)^{\frac{{35}}{4}}}\)
\(A=\dfrac{x^{\dfrac{5}{4}}y+xy^{\dfrac{5}{4}}}{\sqrt[4]{x}+\sqrt[4]{y}}\\ =\dfrac{xy\left(x^{\dfrac{1}{4}}+y^{\dfrac{1}{4}}\right)}{x^{\dfrac{1}{4}}+y^{\dfrac{1}{4}}}\\ =xy\)
\(B=\left(\sqrt[7]{\dfrac{x}{y}\sqrt[5]{\dfrac{y}{x}}}\right)^{\dfrac{35}{4}}\\= \left(\sqrt[7]{\dfrac{x}{y}\cdot\left(\dfrac{x}{y}\right)^{-\dfrac{1}{5}}}\right)^{\dfrac{35}{4}}\\ =\left(\sqrt[7]{\left(\dfrac{x}{y}\right)^{\dfrac{4}{5}}}\right)^{\dfrac{35}{4}}\\ =\left[\left(\dfrac{x}{y}\right)^{\dfrac{4}{35}}\right]^{\dfrac{35}{4}}\\ =\left(\dfrac{x}{y}\right)^{\dfrac{4}{35}\cdot\dfrac{35}{4}}\\ =\left(\dfrac{x}{y}\right)^1\\ =\dfrac{x}{y}\)
35Cho biểu thức
P=\(\left[\left(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}\right)\frac{2}{\sqrt{x}+\sqrt{y}}+\frac{1}{x}+\frac{1}{y}\right]:\frac{\sqrt{x^3}+y\sqrt{x}+x\sqrt{y}+\sqrt{y^3}}{\sqrt{xy^3}+\sqrt{x^3y}}\)
a) Rút gọn P
b)Cho xy=16 . Tìm Min P
34 Cho biểu thức
P=\(\frac{x}{\sqrt{xy}-2y}-\frac{2\sqrt{x}}{x+\sqrt{x}-2\sqrt{xy}-2\sqrt{y}}-\frac{1-x}{1-\sqrt{x}}\)
a) Rút gọn P
b)Tính P biết 2x^2+y^2-4x-2xy+4=0
Rút gọn các biểu thức sau:
a) $\sqrt{9a^4}$
b) 2$\sqrt{a^{2}}$- 5a (với a<0)
c) $\sqrt{16(1+4x+4x^2)}$ với x $\geq$ $\frac{1}{2}$
d) $\frac{1}{a-3}$$\sqrt{9(a^2-3a+9)}$ với a<3
a) \(\sqrt{9a^4}=\sqrt{\left(3a^2\right)^2}=\left|3a^2\right|=3a^2\)
b) \(2\sqrt{a^2}-5a=2\left|a\right|-5a=-2a-5a=-7a\)
c) \(\sqrt{16\left(1+4x+4x^2\right)}=\sqrt{\left[4\left(1+2x\right)\right]^2}=\left|4\left(1+2x\right)\right|=4\left(1+2x\right)\)
Giúp mình với các bạn: Rút gọn biểu thức:
\(\frac{y}{x}.\sqrt{\frac{x^2}{y^2}}\)
với x>0 và y khác 0
\(\frac{y}{x}.\sqrt{\frac{x^2}{y^2}}=\frac{y}{x}.\sqrt{\left(\frac{x}{y}\right)^2}=\frac{y}{x}.\frac{x}{y}=1\)
Chúc bạn học tốt :)
